Journal of Operator Theory

Volume 75, Issue 1, Winter 2016 pp. 91-118.

Diagonality and Idempotents with applications to problems in operator theory and frame theory

Authors: Jireh Loreaux (1) and Gary Weiss (2)
Author institution: (1) Department of Mathematics, McMicken College of Arts and Sciences, University of Cincinnati, Cincinnati, Ohio, 45212, U.S.A.
(2) Department of Mathematics, McMicken College of Arts and Sciences, University of Cincinnati, Cincinnati, Ohio, 45212, U.S.A.

Summary: We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis. We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.

DOI: http://dx.doi.org/10.7900/jot.2014nov05.2054
Keywords: idempotents, diagonals, zero-diagonal, Hilbert-Schmidt perturbation, dual frame

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